Speed of Light - Kat Couteaux

Measurement of the Speed of Light Using a Helium Neon Laser

Kat Couteaux

University of Minnesota, School of Physics and Astronomy

Abstract

The speed of light is a critical constant that has been historically used in astrophysical settings as well as in defining other fundamental constants. The speed of light in air was measured from the beating frequency of a HeNe laser. The beat frequencies were measured over a range of different laser cavity lengths, which are adjusted using a stepper motor. Non-linear effects were minimized by maintaining a constant laser intensity and selecting two longitudinal modes of the same intensity. It was found that the speed of light in air was measured to be (3.0096 +/- 0.0033) x 10^8 m/s, a percent difference of 0.69% from the accepted value of (2.9979 x 10^8) m/s. This result is within 6 standard deviations of the accepted value, which is attributed to non-linear effects, but the precision is still found to be within 1% of the speed of light.

Introduction

Upon entering the late 1600’s, the speed of light was considered to be an infinite quantity. This was until RØmer discovered that the observations of Jupiter’s moon, Io, was dependent on the distance between Earth and the satellite in 1697. However, it was not until 1972 when the speed of light was measured using a methane-stabilized HeNe laser that the accepted value (299,792,458 m/s) was determined. For the purposes of this experiment, we intended to measure the speed of light within 1% of the accepted value. To perform this precision experiment, a HeNe laser in an open resonance cavity bounded by spherical mirrors was used to measure the speed of light. Using this model, only wavelengths of light that are half the length of the cavity length will be able to constructively interfere, each oscillating with a specific resonant frequency. In doing so, we were able to assess these resonant frequencies of light and their resulting beat frequencies in order to determine the speed of light where nonlinear effects were minimized up to 15%.

Theory

Lasing itself occurs when resonant frequencies of light are amplified by stimulated emission inside of a cavity bounded by mirrors. In order to observe this light, one of the mirrors residing on the sides of the cavity is only partially reflective (allowing a small percentage of the light to pass through), as the other is nearly 100% reflective [1]. Stimulated emission is a photon excitation process that occurs when an electron interacts with an incoming photon, emitting two photons with the same momentum as it de-excites into its ground state energy. In doing so, a coherent spectrum of light is emitted. For the purposes of Helium Neon gas (75% Helium and 25% Neon) the light emitted at 633 nm has the highest gain of all other wavelengths emitted. Considering this, this red light dominates over other wavelength in the absence of a filter with some sort of grating.

Only half-wavelengths of light may resonate such that for every Nth longitudinal mode the frequency can be defined as,

Where n is the index of refraction, c is the speed of light, and L is the displaced length of the lasing cavity. At the photodetector, these modes interfere and create beat frequencies, which can be defined as,

Knowing the displacement of the lasing cavity (L), the speed of light is proportional to the linearity between the displacement of the cavity and inverse of beat frequency (∆f) such that,

Where L_0 is a constant. Using this relationship, the speed of light can be calculated from the resulting slope.

Methods

Light resonates inside of a lasing cavity bounded by a high reflector with a radius of curvature of 0.045 m at one end of the laser tube and a 99% reflective spherical mirror (output coupler) with radius of curvature of 0.06 m on a translation stage (stepper motor). The resulting resonant frequencies are detected by a scanning Fabry-Perot and their beat frequencies are detected by a photodetector. The output of the scanning Fraby-Perot is read by the oscilloscope and the output of the photodetector is read by the spectrum analyzer. The laser consists of a 28 cm glass tube filled with HeNe gas. The stepper motor allows for changes in the length of the laser cavity with a precision of 1/402.2 mm. All measurements were recorded using a LabVIEW program connected to the two detectors.

Results

The data taken represents measurements of beat frequency taken over 16 different cavity lengths between 0 and 2 mm. Our final results are displayed above, where the speed of light was calculated to be c = (3.0096 +/- 0.0033 x 10^8) m/s. Here, the uncertainty is fixed for measurements of length. However, when taking the inverse of the beat frequency, this is not necessarily true. By minimizing our chi^2 value, the uncertainty in the beat frequency can be calculated to be (+/- 1.0800 x 10^4) Hz.

By creating a linear relationship between the number of steps and the displacement of the stepper motor it was found that 1 mm is traveled for every 402.2 steps. The uncertainty in the displacement of the cavity was (+/- 1.2300 x 10^-6) mm. This is derived from the fact that measured values deviated from our expected value of 1/400 mm/steps (based on the program constructed in LabVIEW) by +/- 2 steps.

Further, we can evaluate the goodness of fit of the linear relationship between cavity displacement and the inverse of beat frequency. Here, it is apparent that the majority of data (about 2/3) reside within +/- 1 of the weighted residuals, implying the least square fitting yielded an appropriate fit for this set of data. Considering this, the uncertainties propagated for both cavity displacement and beat frequencies are also appropriate.

It becomes apparent that the measured value is still 6 standard deviations away from the accepted value. Considering the measurements from the stepper motor did not deviate dramatically from the predicted value (1/400 mm/steps), the discrepancies in the speed of light may be attributed to frequency pushing. Though the intensities of resonant frequencies were kept within 15% of each other, fluctuations in the intensities resulted in deviations in beat frequencies up to 10 kHz. Considering this, the deviation of the measured value to the accepted value of the speed of light can be attributed to non-linear effects in the beat frequency.

Conclusion

This experiment not only explored the fundamentals of lasers in optics, but exploited the nature of lasing to derive a fundamental constant. Without the speed of light we would not stand where we are technologically, scientifically, and culturally. We would not have the precision to evaluate properties of distant stars and galaxies, nor the precision to define other fundamental constants. Essentially, understanding and building a consensus on the speed of light is a crucial component in modern science.

Acknowledgements

I extend thanks to Kurt Wick for advising this project, and to Kevin Booth and Professor Roger Rusak for additional academic support.

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